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In Exercises polarcompbasicfirst - polarcompbasiclast, find a polar representation for the complex number \(z\). Identify \(\text {Re}(z)\), \(\text {Im}(z)\), \(|z|\), \(\text {arg}(z)\) and \(\text {Arg}(z)\).
In Exercises rectcompfirst - rectcomplast, find the rectangular form of the given complex number. Use whatever identities are necessary to find the
exact values.
For Exercises polarcomparithfirst - polarcomparithlast, use \(z = -\frac {3\sqrt {3}}{2} + \frac {3}{2}i\) and \(w = 3\sqrt {2} - 3i\sqrt {2}\) to compute the quantity. Express your answers in polar form using the principal
argument.
In Exercises demoivrefirst - demoivrelast, use DeMoivre’s Theorem to find the indicated power of the given complex number. Express your final
answers in rectangular form.
In Exercises polarrootsfirst - polarrootslast, find the indicated complex roots. Express your answers in polar form and then convert them into rectangular
form.
the two square roots of \(z = 4i\)
Since \(z=4i = 4\text {cis}\left (\frac {\pi }{2}\right )\) we have
Use the Sum and Difference Identities in Theorem circularsumdifference or the Half Angle Identities in Theorem halfangle to convert the three cube roots
of \(z=\sqrt {2} + i\sqrt {2}\) we found in Example nthrootscomplexex, number halfanglecuberoot from polar form to rectangular form.
Note: In the answers for \(w_{\text {\tiny $0$}}\) and \(w_{\text {\tiny $2$}}\) the first rectangular form comes from applying the appropriate Sum or Difference Identity (\(\frac {\pi }{12} = \frac {\pi }{3} - \frac {\pi }{4}\) and \(\frac {17\pi }{12} = \frac {2\pi }{3} + \frac {3\pi }{4}\),
respectively) and the second comes from using the Half-Angle Identities.
Use a calculator to approximate the rectangular form of the five fifth roots of \(1\) we found in Example nthrootscomplexex, number
calculatorfifthroot.
According to Theorem realfactorization in Section ComplexZeros, the polynomial \(p(x) = x^{4} + 4\) can be factored into the product linear and irreducible quadratic factors.
In Exercise factorpolywithnonlinear in Section NonLinearEquations, we showed you how to factor this polynomial into the product of two irreducible quadratic factors
using a system of non-linear equations. Now that we can compute the complex fourth roots of \(-4\) directly using Theorem nthrootscomplexthm, we
can apply the Complex Factorization Theorem, Theorem complexfactorization, to obtain the linear factorization \(p(x) = (x - (1 + i))(x - (1 - i))(x - (-1 + i))(x - (-1 - i))\). By multiplying the first two factors
together and then the second two factors together, thus pairing up the complex conjugate pairs of zeros Theorem conjugatepairsthm told us
we’d get, we have that \(p(x) = (x^{2} - 2x + 2)(x^{2} + 2x + 2)\). Use the 12 complex \(12^{\text {th}}\) roots of 4096 to factor \(p(x) = x^{12} - 4096\) into a product of linear and irreducible quadratic
factors.
Use Exercise triangleineqforvectorsexercise from Section TheDotProduct to show the the Triangle Inequality \(|z + w| \leq |z| + |w|\) holds for all complex numbers \(z\) and \(w\) as well. Identify the
complex number \(z = a + bi\) with the vector \(u = \langle a, b \rangle \) and identify the complex number \(w = c + di\) with the vector \(v = \langle c, d \rangle \) and just follow your nose!
Complete the proof of Theorem modprops by showing that if \(w \neq 0\) than \(\left | \frac {1}{w}\right | = \frac {1}{|w|}\).
Recall from Section ComplexZeros that given a complex number \(z = a+bi\) its complex conjugate, denoted \(\overline {z}\), is given by \(\overline {z} = a - bi\).
Prove that \(\left | \overline {z} \right | = |z|\).
Prove that \(|z| = \sqrt {z \overline {z}}\)
Show that \(\text {Re}(z) = \frac {z + \overline {z}}{2}\) and \(\text {Im}(z) = \frac {z - \overline {z}}{2i}\)
Show that if \(\theta \in \text {arg}(z)\) then \(-\theta \in \text {arg}\left (\overline {z}\right )\). Interpret this result geometrically.
Is it always true that \(\text {Arg}\left (\overline {z}\right ) = -\text {Arg}(z)\)?
Given a natural number \(n \geq 2\), the \(n\) complex \(n^{\text {th}}\) roots of \(z = 1\) are called the \(n^{\mbox {\textbf {\scriptsize th}}}\) Roots of Unity. In the following exercises, assume that \(n\) is a
fixed, but arbitrary, natural number such that \(n \geq 2\).
Show that \(w = 1\) is an \(n^{\text {th}}\) root of unity.
Show that if both \(w_{\text {\tiny $j$}}\) and \(w_{\text {\tiny $k$}}\) are \(n^{\text {th}}\) roots of unity then so is their product \(w_{\text {\tiny $j$}}w_{\text {\tiny $k$}}\).
Show that if \(w_{\text {\tiny $j$}}\) is an \(n^{\text {th}}\) root of unity then there is an \(n^{\text {th}}\) root of unity \(w_{\text {\tiny $j'$}}\) so that \(w_{\text {\tiny $j$}}w_{\text {\tiny $j'$}} = 1\).
HINT: If \(w_{\text {\tiny $j$}} = \text {cis}(\theta )\) let \(w_{\text {\tiny $j'$}} = \text {cis}(2\pi - \theta )\). Show \(w_{\text {\tiny $j'$}} = \text {cis}(2\pi - \theta )\) is indeed an \(n^{\text {th}}\) root of unity.
Another way to express the polar form of a complex number is to use the exponential function. For real numbers \(t\), Euler’s
Formula defines \(e^{it} = \cos (t) + i \sin (t)\).
\(e^{ix} e^{iy} = e^{i(x+y)}\) for all real numbers \(x\) and \(y\).
\(\left (e^{ix}\right )^{n} = e^{i(nx)}\) for any real number \(x\) and any natural number \(n\).
\(\frac {e^{ix}}{e^{iy}} = e^{i(x-y)}\) for all real numbers \(x\) and \(y\).
If \(z = r\text {cis}(\theta )\) is the polar form of \(z\), show that \(z = re^{it}\) where \(\theta = t\) radians.
Show that \(e^{i\pi } + 1 = 0\). (This famous equation relates the five most important constants in all of Mathematics with the three most
fundamental operations in Mathematics.)
Show that \(\cos (t) = \frac {e^{it} + e^{-it}}{2}\) and that \(\sin (t) = \frac {e^{it} - e^{-it}}{2i}\) for all real numbers \(t\).
